First off, if someone flat out does not like anime with all their soul, it will be nearly impossible to make them like it; don’t bother trying to force your hobby on them. Instead, help the people who might have a skewed interpretation of the subject, like anime is for kids only, or anime is all about tentacle rape, or anime is only Pokemon. As with anything in this world, first impressions mean everything. They will judge the entire anime race based on the first few minutes of a movie or a television show, so it has to be a good one. Depending on how often you see this person, and how committed they are at trying something new a try will tell you whether to watch a movie, or start a series.
Ghibli film is always a safe bet. But sometimes, it can be too safe because of it’s similarities to Disney films. You want to showcase how anime is different than western animation, so be sure to pick the right title. For example, if they truly believe that anything animated is for children, then show them Grave of the Fireflies. This movie is about the harshness of war on both sides of the battlefield and the sadness that accompanies war, it will definitely prove your point. Also most Satoshi Kon films will be able to showcase the more serious side of anime as well.
If a friend or a family member is in this for the long run, then a series will be the best choice. Anime series are the primary way most people watch anime, and in my opinion, the best way. The number one rule to follow in this situation is to pick a short series that matches the viewer’s taste. Cowboy Bebop is a series that will impress the most despite what their taste might be. I’ve also started people on shows like Naruto or One Piece in the past, but it retrospect it might not have been the best idea. Most of us like to start what we finish, and since these two shows will never end, and take an enormous time to finish, your friends and family will spend all their times watching these shows and not experimenting with others.
Like I said before, the most important thing you can do to keep them watching anime is to make a good first impression. There is nothing different between people who watch anime, and those that don’t, other than a bad experience in the past that kept people from watching it. Reeducate them one step at a time, and those people will be thanking you in the future.